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      Solving complex linear equations

      Lesson details

      Learning outcome

      I can solve complex linear equations, including those involving reciprocals.

      Key learning points

      1. Any linear equation written in the form Ax + B = C can be solved.
      2. Equations involving division by a constant can be manipulated to reach the form Ax + B = C.
      3. Equations involving division by a variable can be manipulated to reach the form Ax + B = C.

      Keywords

      • Equation - An equation is used to show two expressions that are equal to each other.

      Common misconception

      Pupils often "multiply through by $$5$$" and turn $${{2x+1}\over5}={10}$$ into $${{10x+5}}={50}$$

      Use equivalent fractions as opposed to multiplying through. In one simple step $${{2x+1}\over5}={10}$$ becomes $${{2x+1}\over5}={50\over5}$$

      Teacher tip

      The key is for pupils to be able to fluently use their fractions skills and knowledge when manipulating algebra. Use a numerical fractions starter to ensure they recall key skills like multiplying a fraction by an integer, and by another fraction.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      $$1\over7$$ is the of $$7$$

      fraction
      division
      Correct answer: reciprocal
      divisor
      multiplier

      Q2.
      Solve $$6x+9=27$$

      Correct answer: $$x=3$$
      $$x=6$$
      $$x=9$$
      $$x=18$$

      Q3.
      Solve $${x\over3}=15$$

      Correct answer: $$x=45$$
      $$x=18$$
      $$x=5$$
      $$x={1\over5}$$

      Q4.
      Calculate $${2\over3}\times{4\over5}$$

      $$6\over15$$
      $$6\over20$$
      Correct answer: $$8\over15$$
      $$10\over12$$
      $$12\over10$$

      Q5.
      Which of these expressions are equivalent to $${{x+1}\over3}\times{2\over3}$$ ?

      $${2x+1}\over9$$
      Correct answer: $${2{(x+1)}\over9}$$
      $${2{(x+1)}\over3}$$
      $${2x+2}\over3$$
      Correct answer: $${2x+2}\over9$$

      Q6.
      Expand and simplify $$8({{x}\over8})$$

      $$8$$
      Correct answer: $$x$$
      $$8x$$
      $${{8x}\over8}$$
      $$x=8$$

      6 Questions

      Q1.
      $${{56}\over{y}}=7$$ is an example of a equation.

      variable
      Correct answer: rational
      irrational
      special

      Q2.
      The solution to $${{6x−2}\over{4}}=7$$ is $$x=$$

      Correct Answer: 5, five

      Q3.
      Solve $${{2x}\over{3}}+1=11$$

      Correct answer: $$x=15$$
      $$x=16$$
      $$x=17$$
      $$x=18$$

      Q4.
      Which is the most efficient first step when solving the equation $${{x+7}\over{9}}={{x-1}\over{3}}$$ ?

      Subtract $$7$$ from both sides.
      Add $$1$$ to both sides.
      Correct answer: Multiply both sides by $$9$$
      Multiply both sides by $$3$$
      Multiply both sides by $$27$$

      Q5.
      Solve $${{x+7}\over{9}}={{x−1}\over{3}}$$. The solution is $$x=$$

      Correct Answer: 5, five

      Q6.
      Solve $${{32}\over{3x}}=8$$

      $$x={3\over4}$$
      $$x=3$$
      $$x=4$$
      Correct answer: $$x={4\over3}$$

      To help you plan your 8 maths lesson on: Solving complex linear equations, download all teaching resources for free and adapt to suit your pupils' needs...